Theorem rossspw 31421

Description: A ring of sets is a collection of subsets of 𝑂. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
isros.1	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}

Assertion

Ref	Expression
rossspw	⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦

Allowed substitution hints:   𝑄(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦)

Proof of Theorem rossspw

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isros.1	. . . 4 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
2	1	isros 31420	. . 3 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
3	2	simp1bi 1140	. 2 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
4	3	elpwid 4551	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  {crab 3140   ∖ cdif 3931   ∪ cun 3932   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pw 4539

This theorem is referenced by:  rossros  31432